where \(W\)  is a Brownian motion. The main reason is the Itô's integral \(\int _t^{t+\epsilon }\sigma (s,x_s^u,u_s)dW_s\)  is only of order \(O(\epsilon ^{\frac{1}{2}})\) . In 1990, Peng [1]} first introduced the second-order term in the Taylor's expansion of the variation and completely obtained the global maximum principle for the stochastic control problem.
